Why is the property when you multiply two modulars (you multiply the two ones on outside and the two ones inside)
Why does that property hold true?
Addition is easy but multiplication doesn't make sense to me conceptually.
I can't seem to find any intuition behind the multiplication rule. Thank you!
Product Rule: $X \equiv x \text{ mod } m$ and $Y \equiv y \text{ mod } m \implies X \cdot Y \equiv x \cdot y \text{ mod } m$.
Proof: See here. Another way is to simply expand out the terms, do the calculations using normal arithmetic, and then express the result back in modular arithmetic.
Expansion: $\begin{align}X &\equiv x \text{ mod } m \iff X = k_1 \cdot m + x \text{ where } k_1 \in \mathbb{Z} \\ Y &\equiv y \text{ mod } m \iff Y = k_2 \cdot m + y \text{ where } k_2 \in \mathbb{Z} \end{align}$
Calculation using normal arithmetic:
$\begin{align}X \cdot Y &= (k_1 \cdot m + x) \cdot (k_2 \cdot m + y)\\ &=\underbrace{(k_1k_2m + k_1y + k_2x)}_{k \in \mathbb{Z}}\cdot m + xy \\ &= k \cdot m + xy\end{align}$
Result: $X \cdot Y = k \cdot m + xy \iff X \cdot Y \equiv xy \text{ mod } m$
Conceptually, multiplication between two integers $X, Y$ can be thought of as repeated addition (assuming $Y > 0$), so it's not surprising that a property of integer addition (congruence sum rule) applied repeatedly may lead to a property of integer multiplication.
mod m: $\begin{align}X \cdot Y &\equiv \underbrace{X + X + X + \cdots + X}_\text{Y times}\\& \equiv \underbrace{x + x + x + \cdots + x}_\text{Y times} &\text{sum rule applied Y-1 times}\\ &\equiv x \cdot Y = Y \cdot x \\&\equiv \underbrace{Y + Y + Y + \cdots + Y}_\text{x times}\\& \equiv \underbrace{y + y + y + \cdots + y}_\text{x times} &\text{sum rule applied x-1 times}\\ &\equiv yx = xy\end{align}$
